Theorem uvtx01vtx 26020

Description: If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated ((𝑉 UnivVertex ∅) ≠ ∅ ↔ (#‘𝑉) = 1), but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)

Assertion

Ref	Expression
uvtx01vtx	⊢ ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥})

Distinct variable group:   𝑥,𝑉

Proof of Theorem uvtx01vtx

Dummy variables 𝑒 𝑘 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4718	. . . . 5 ⊢ ∅ ∈ V
2		isuvtx 26016	. . . . 5 ⊢ ((𝑉 ∈ V ∧ ∅ ∈ V) → (𝑉 UnivVertex ∅) = {𝑥 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅})
3	1, 2	mpan2 703	. . . 4 ⊢ (𝑉 ∈ V → (𝑉 UnivVertex ∅) = {𝑥 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅})
4	3	neeq1d 2841	. . 3 ⊢ (𝑉 ∈ V → ((𝑉 UnivVertex ∅) ≠ ∅ ↔ {𝑥 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅} ≠ ∅))
5		rabn0 3912	. . . 4 ⊢ ({𝑥 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅} ≠ ∅ ↔ ∃𝑥 ∈ 𝑉 ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅)
6	5	a1i 11	. . 3 ⊢ (𝑉 ∈ V → ({𝑥 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅} ≠ ∅ ↔ ∃𝑥 ∈ 𝑉 ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅))
7		df-rex 2902	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅ ↔ ∃𝑥(𝑥 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅))
8		noel 3878	. . . . . . . . . 10 ⊢ ¬ {𝑘, 𝑥} ∈ ∅
9		rn0 5298	. . . . . . . . . . 11 ⊢ ran ∅ = ∅
10	9	eleq2i 2680	. . . . . . . . . 10 ⊢ ({𝑘, 𝑥} ∈ ran ∅ ↔ {𝑘, 𝑥} ∈ ∅)
11	8, 10	mtbir 312	. . . . . . . . 9 ⊢ ¬ {𝑘, 𝑥} ∈ ran ∅
12	11	ralf0 4030	. . . . . . . 8 ⊢ (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅ ↔ (𝑉 ∖ {𝑥}) = ∅)
13	12	a1i 11	. . . . . . 7 ⊢ (𝑉 ∈ V → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅ ↔ (𝑉 ∖ {𝑥}) = ∅))
14	13	anbi2d 736	. . . . . 6 ⊢ (𝑉 ∈ V → ((𝑥 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅) ↔ (𝑥 ∈ 𝑉 ∧ (𝑉 ∖ {𝑥}) = ∅)))
15		ssdif0 3896	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑥} ↔ (𝑉 ∖ {𝑥}) = ∅)
16	15	a1i 11	. . . . . . . 8 ⊢ (𝑉 ∈ V → (𝑉 ⊆ {𝑥} ↔ (𝑉 ∖ {𝑥}) = ∅))
17	16	bicomd 212	. . . . . . 7 ⊢ (𝑉 ∈ V → ((𝑉 ∖ {𝑥}) = ∅ ↔ 𝑉 ⊆ {𝑥}))
18	17	anbi2d 736	. . . . . 6 ⊢ (𝑉 ∈ V → ((𝑥 ∈ 𝑉 ∧ (𝑉 ∖ {𝑥}) = ∅) ↔ (𝑥 ∈ 𝑉 ∧ 𝑉 ⊆ {𝑥})))
19		sssn 4298	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑥} ↔ (𝑉 = ∅ ∨ 𝑉 = {𝑥}))
20	19	anbi2i 726	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ⊆ {𝑥}) ↔ (𝑥 ∈ 𝑉 ∧ (𝑉 = ∅ ∨ 𝑉 = {𝑥})))
21		n0i 3879	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑉 → ¬ 𝑉 = ∅)
22	21	pm2.21d 117	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑉 → (𝑉 = ∅ → 𝑉 = {𝑥}))
23	22	imp 444	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 = ∅) → 𝑉 = {𝑥})
24		simpr 476	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 = {𝑥}) → 𝑉 = {𝑥})
25	23, 24	jaodan 822	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑉 = ∅ ∨ 𝑉 = {𝑥})) → 𝑉 = {𝑥})
26		vsnid 4156	. . . . . . . . . . 11 ⊢ 𝑥 ∈ {𝑥}
27		eleq2 2677	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑥} → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ {𝑥}))
28	26, 27	mpbiri 247	. . . . . . . . . 10 ⊢ (𝑉 = {𝑥} → 𝑥 ∈ 𝑉)
29		olc 398	. . . . . . . . . 10 ⊢ (𝑉 = {𝑥} → (𝑉 = ∅ ∨ 𝑉 = {𝑥}))
30	28, 29	jca 553	. . . . . . . . 9 ⊢ (𝑉 = {𝑥} → (𝑥 ∈ 𝑉 ∧ (𝑉 = ∅ ∨ 𝑉 = {𝑥})))
31	25, 30	impbii 198	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑉 = ∅ ∨ 𝑉 = {𝑥})) ↔ 𝑉 = {𝑥})
32	20, 31	bitri 263	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ⊆ {𝑥}) ↔ 𝑉 = {𝑥})
33	32	a1i 11	. . . . . 6 ⊢ (𝑉 ∈ V → ((𝑥 ∈ 𝑉 ∧ 𝑉 ⊆ {𝑥}) ↔ 𝑉 = {𝑥}))
34	14, 18, 33	3bitrd 293	. . . . 5 ⊢ (𝑉 ∈ V → ((𝑥 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅) ↔ 𝑉 = {𝑥}))
35	34	exbidv 1837	. . . 4 ⊢ (𝑉 ∈ V → (∃𝑥(𝑥 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅) ↔ ∃𝑥 𝑉 = {𝑥}))
36	7, 35	syl5bb 271	. . 3 ⊢ (𝑉 ∈ V → (∃𝑥 ∈ 𝑉 ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅ ↔ ∃𝑥 𝑉 = {𝑥}))
37	4, 6, 36	3bitrd 293	. 2 ⊢ (𝑉 ∈ V → ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥}))
38		id 22	. . . . . . 7 ⊢ (¬ 𝑉 ∈ V → ¬ 𝑉 ∈ V)
39	38	intnanrd 954	. . . . . 6 ⊢ (¬ 𝑉 ∈ V → ¬ (𝑉 ∈ V ∧ ∅ ∈ V))
40		df-uvtx 25951	. . . . . . 7 ⊢ UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛 ∈ 𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})
41	40	mpt2ndm0 6773	. . . . . 6 ⊢ (¬ (𝑉 ∈ V ∧ ∅ ∈ V) → (𝑉 UnivVertex ∅) = ∅)
42	39, 41	syl 17	. . . . 5 ⊢ (¬ 𝑉 ∈ V → (𝑉 UnivVertex ∅) = ∅)
43	42	notnotd 137	. . . 4 ⊢ (¬ 𝑉 ∈ V → ¬ ¬ (𝑉 UnivVertex ∅) = ∅)
44		df-ne 2782	. . . 4 ⊢ ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ¬ (𝑉 UnivVertex ∅) = ∅)
45	43, 44	sylnibr 318	. . 3 ⊢ (¬ 𝑉 ∈ V → ¬ (𝑉 UnivVertex ∅) ≠ ∅)
46		snex 4835	. . . . . 6 ⊢ {𝑥} ∈ V
47		eleq1 2676	. . . . . 6 ⊢ (𝑉 = {𝑥} → (𝑉 ∈ V ↔ {𝑥} ∈ V))
48	46, 47	mpbiri 247	. . . . 5 ⊢ (𝑉 = {𝑥} → 𝑉 ∈ V)
49	48	exlimiv 1845	. . . 4 ⊢ (∃𝑥 𝑉 = {𝑥} → 𝑉 ∈ V)
50	49	con3i 149	. . 3 ⊢ (¬ 𝑉 ∈ V → ¬ ∃𝑥 𝑉 = {𝑥})
51	45, 50	2falsed 365	. 2 ⊢ (¬ 𝑉 ∈ V → ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥}))
52	37, 51	pm2.61i 175	1 ⊢ ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥})

Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ran crn 5039  (class class class)co 6549   UnivVertex cuvtx 25948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-uvtx 25951

This theorem is referenced by: (None)